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"What's in a triangle? That which we call a three-sided polygon By any other name would look as such." –Euclid, Geomet-Romeo and Juli-Ometry

In this particular quote, Euclid (who wrote plays about shapes in Elizabethan English when he wasn't studying geometry) asks what makes the name "triangle" so special. After all, we can call a triangle anything we want and it would still be the same shape, right?

While that's true, these particular polygons are called triangles for very good reasons. If we split up the name into "tri-" and "-angle," the meaning of the word becomes very clear. The "tri-" is for three (like tripod or tricycle) and "-angle" is for, well, angle.

Example 1

What is the measure of ∠DFE? What sort of triangle is ∆DEF?

From the picture, we can see that the three sides of the triangle, DF, FE, and ED are all congruent and therefore have the same length. All the angles are congruent as well. That makes ∆DEF an equilateral triangle.

How can we figure out the angles of an equilateral triangle? Spoiler alert: They're all equal. We'll call each angle measurement x. Since they're all equal and we know all of a triangle's angles add up to 180°, we can set up this equation.

x + x + x = 3x = 180x = 60

In other words, all the angles in an equilateral triangle have measurements of 60°, including our dear ∠DFE.

Example 3

For this problem, we don't have exterior angles to help us out. All we know is that all the angles add up to 180°. We gotta start somewhere.

180 = m∠CHG + m∠HGC + m∠GCH

Luckily, m∠CHG = m∠HGC, so we can replace them both with another term. The letter x is too common, so let's use q. It's quacky and quivering with quirkiness.

180 = 2q + m∠GCH

What about ∠GCH? How are we supposed to figure out its measure? We should be careful not to guess too high a number. We don't want it to think we're calling it fat.

Actually, we can figure out its exact measure. Since ∠GCH and ∠BCD are vertical angles, they're congruent and have the same measure. That means m∠GCH = 64. We're safe, since 64° is cute and acute, much like saying, "Of course you aren't fat, honey. You're the perfect size."

180 = 2q + 64q = 58

That means m∠CHG = 58.

What about the type of triangle? If ∠CHG and ∠HGC being congruent weren't enough, CH and CG are congruent as well. That's a clear-cut isosceles. Pair that up with its three acute angles, and ∆CHG is an acute isosceles triangle.

Exercise 1

Use the triangle Angle Sum Theorem. Or try fitting two elephants into a one-bedroom flat.

Answer

Let's see what happens if we say that ∠1, ∠2, and ∠3 make up the interior angles of a triangle and both ∠1 and ∠2 are obtuse. Obtuse angles are greater than 90°, so just adding the measures of ∠1 and ∠2 would be greater than 180. That means ∠3 would have to be negative and we can't have negative angles.